Barycentric Dynamical Time for the HP-41

Overview
 

 1°)  Simplified Formula
 2°)  INPOP10
 3°)  Barycentric Coordinate Time
 

-Several time scales are available in astronomical work:
-The Terrestrial Time = TT  which is related to the International Atomic Time TAI  by  TT = TAI + 32.184 seconds.

-The Barycentric Dynamical Time = TDB = Temps Dynamique Barycentrique which is preferable for the motions of planets in the Solar system.
-Due to relativistic effects, there is a difference between TDB & TT that the following programs evaluate.
-This difference remains small: less than 2 milliseconds, at least several millennia around J2000.
 

1°)  Simplified Formula
 

-Reference [1] gives a very simple formula:

   TDB - TT = 0.001658 sin M + 0.000014 sin 2M    where   M is the Sun's mean anomaly.

-But the program hereunder employs the first terms of the series given in reference [2] or [3],
  those whose amplitude > 0.4 second over the time-span [1000,3000]
-The errors should not exceed 1 or 2 microseconds over this interval of time.
 

Data Registers:   R00 thru R03: temp   ( R00 = number of millennia since 2000/01/01  12h )
Flags: /
Subroutine:    "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )
 
 

 
    ( 259 bytes / SIZE 004 )
 
 

 
Example1:      2012/08/29  16h41m37s  ( TT or TDB )

    2012.0829    ENTER^
        16.4137    XEQ "TDB"  >>>>   TDB-TT =  - 0.0013227 s               ---Execution time =18s---
 

Example2:      3000/04/04  1h23m45s  ( TT or TDB )

    3000.0404   ENTER^
          1.2345      R/S        >>>>    TDB-TT = + 0.0015386 s
 

Note:

-In reference [1], the inputs should be expressed in TDB whereas in refence [2], they should be expressed in TT.
-To the level of accuracy of this simplified formula, the difference is negligible.
 

2°)  INPOP10
 

-Like the JPL ephemerides DE4xx, the French ephemerides INPOP ( Integrations Numériques Planétaires de l'Observatoire de Paris )
 give the positions and velocities of the bodies of the Solar System under the form of Chebyshev Polynomials.
-But instead of the nutation, the difference TT-TDB is also provided: 12 coefficients for each time-span of 4 days, cf reference [4]
 
 

Data Registers:        •  R00 = bbb.eee               ( Registers R00 & Rbb thru Ree are to be initialized before executing "TDB2" )

                                   •  Rbb = JD1   •  Rbb+1 = JD2      •  Rbb+2 = a₀    •  Rbb+3 = a₁  ...............................   •  Ree = a_n
Flags: /
Subroutines:      "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )
                             "CdT"  ( cf "Orthogonal Polynomials §5-b) or "Ephemerides and Chebyshev Polynomials for the HP-41" )
 
 

 
( 59 bytes / SIZE ??? )
 
 

 
Example:       3000/04/04  1h23m45s

-In reference [4], we find a text file that contains the following constants:

      2816877.00         1st  Julian Date = 3000/03/31  12h  TDB
      2816881.00         2nd Julian Date = 3000/04/04  12h  TDB

     -0.15254574201273665E-02
     -0.17329122056166975E-04
     +0.52937767551143974E-06
     +0.27550591219803854E-08
     -0.34148361610150929E-09
     -0.69921867908556206E-11
    +0.85016053321457848E-12
    +0.21451895272947141E-14
     -0.14234824035252474E-14
    +0.17399035530231268E-15
     -0.19832610480477758E-17
     -0.14385810488591148E-17

-After storing these 14 coefficients in, say R01 to R14  ( control number = 1.014 )

    1.014  STO 00

    3000.0404   ENTER^
          1.2345   XEQ "TDB2"  >>>>    TDB-TT = + 0.001538845852 s

Notes:

-The argument of the INPOP ephemeris is TDB ( you can also choose TCB ), so if the input time is expressed in TT,
  we must execute "TDB2" again with Time = 1h32m45s + (TDB-TT) above to get the correct result
-In this example, it yields  TDB-TT = + 0.001538845852 s

-Here, the result is unchanged and it will be almost always the same because the HP-41 works with 10 digits.
-Anyway, one iteration is always sufficient.
 

3°)  Barycentric Coordinate Time
 

-Anothe time scale may also be used:  TCB = Temps Coordonné Barycentrique = Barycentric Coordinate Time.
-Whereas  TDB - TT  remains smaller than 0.002 second at least several millenia aroud J2000, the difference  TCB - TDB  is much larger:
-It was almost 0 in 1977 but in 2012 , TCB -TDB is about 17 seconds.

Formula:

   TCB = TDB + 1.550519768 E-8 ( JD_TCB - 2443144.5003725 ) 86400 + 6.55 E-5 seconds

     where  JD_TCB  is the Julian Date in the TCB scale.

-The program below uses an iteration to compute  TCB - TDB  for a given date expressed in  TDB
 
 

Data Registers:   R00-R01:  temp
Flags: /
Subroutine:     "J0" or "J1" or J2"  ( cf "Julian & Gregorian Calendars for the HP-41" )
 
 

 
( 72 bytes / SIZE 002 )
 
 

  Where the input time is expressed in TDB

Example:       3000/04/04  1h23m45s  TDB

     3000.0404   ENTER^
           1.2345   XEQ "TCB"   >>>>   500.6752393 seconds
 
 

References:

[1]  Robin M. Green - "Spherical Astronomy" - Cambridge University Press - ISBN  0-521-31779-7
[2] "Introduction aux Ephemerides Astronomiques" - EDP Sciences - ISBN 2-86883-298-9  ( in French )
[3]  Fairhead & Bretagnon - "An Analytica Formula for the Time Transformation TB-TT" Astronomy & Astrophysics 229, 240-247 ( 1990 )
[4]  http://www.imcce.fr/inpop
[5]  IAU 2006 Resolution B3